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On the measure theoretic structure of compact groups

Authors:	S Grekas S Mercourakis

Institution:	Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece ; Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece

Abstract:	If is a compact group with $w(G)=a\geq \omega$ , we show the following results: (i) There exist direct products $\displaystyle{\prod _{\xi<a}G_{\xi}, \prod _{\xi<a}H_{\xi}}$ of compact metric groups and continuous open surjections $\displaystyle{\prod _{\xi<a}G_{\xi} \stackrel{p}{\rightarrow }G \stackrel{q}{\rightarrow }\prod _{\xi<a}H_{\xi}}$ with respect to Haar measure; and (ii) the Haar measure on is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups. Applications of the above results in measure theory are given.

Keywords:	Compact group Haar measure Baire isomorphism Riemann integrable function Jordan measurable set

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